An Epistemological and Didactic Study of a Specific Calculus Reasoning Rule

Abstract It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind “for all X, there exists Y such that R(X, Y)”, and forgets that in that case, a priori, “Y depends on X”. We analyse this mistake from both a logical and mathematical point of view, and study it through two inquiries, an historical one and a didactic one. We show that mathematics teachers emphasise the importance of the dependence rule in order to avoid this kind of mistake, while natural deduction in predicate calculus provides a logical framework to analyse and control the use of quantifiers. We show that the relevance of this dependence rule depends heavily on the context: nearly without interest in geometry, but fundamental in analysis or linear algebra. As a consequence, mathematical knowledge is a key to correct reasoning, so that there is a large distance between beginners' and experts' abilities regarding control of validity, that, to be shortened, probably requires more than a syntactic rule or informal advice. Résumé Les difficultés de manipulation, par les étudiants, des énoncés contenant deux quantificateurs différents, rencontrés dans de nombreux raisonnements en analyse, sont bien attestées. Nous nous intéressons plus spécialement dans cet article à une erreur qui appara?t dans certaines preuves lorsque l'on applique deux fois ou plus un énoncé de la forme “pour tout X, il existe Y tel que R(X,Y)” et que l'on oublie que dans un tel cas, a priori, “Y dépend de X”. Nous analysons cette erreur d'un point de vue logique et d'un point de vue mathématique, puis nous l'étudions à travers deux enquêtes, l'une historique et l'autre didactique. Nous montrons que les professeurs de mathématiques soulignent l'importance de la règle de dépendance pour éviter ce type d'erreur, tandis que la déduction naturelle dans le calcul des prédicats fournit un cadre de référence logique pour analyser et contr?ler l'usage des quantificateurs. Nous montrons que la pertinence de la règle de dépendance dépend fortement du contexte: pratiquement sans intérêt en géométrie, elle est tout à fait fondamentale en analyse et en algèbre linéaire. De ce fait, les connaissances mathématiques sont la clé d'un raisonnement correct, si bien qu'il y a une grande distance entre le débutant et l'expert concernant le contr?le de la validité, que quelques règles syntaxiques ou quelques conseils informels ne permettent vraisemblablement pas de réduire.     

Social studies textbooks in the 1990s

Despite continuing criticism, textbooks remain essential to the teaching of social studies in elementary and high schools. The problems laid at their doorstep are, in fact, reflections of the nation’s educational difficulties. As the number of publishers in the textbook market decreases, the market itself becomes less uniform, and the arguments surrounding content become louder, quality may decline In addition, the use of trade and supplementary materials may increase, niche textbooks may multiply, and CD-ROM may drive even the best textbooks into extinction.     

The electronic alternative: Sociological Research Online

Examines how the electronic journal Sociological Research Online has rapidly established itself as a leading journal and how it has gained credibility; discusses possible future sources of revenue.     

Producing Scientific Explanations in Physics—a Multimodal Account

Research in Science Education - This study focuses on the relationship between students’ use of language resources and their conceptualisation of phenomena in producing scientific...     

‘Role stretch’: assessing the blurring of teaching and non‐teaching in the classroom assistant role in Scotland

With the increasing number of teaching and classroom assistants across the UK there is now much debate about what their role should be. In particular concerns have arisen about the extent to which they overstep the boundary from supporting teaching and learning into teaching pupils. This study assesses this issue within Scotland. It draws on a national survey of 2000 head teachers, teachers and classroom assistants and interviews with directors of education in nearly half of all Scottish local authorities. Findings from the research suggest a small number of classroom assistants in Scotland are overstepping the boundary into teaching. The paper concludes with an explanation as to why this is happening taking into account aspects such as local authority policy, school size and the individual characteristics of the classroom assistants.     

Student perceptions of the classroom environment: Relations to motivation and achievement in mathematics

This study examined the relationship of 979 middle school students’ perceptions of their mathematics classroom environment to their motivation and achievement. Structural equation modeling indicated that motivational variables (utility, personal achievement goals, efficacy) mediated the influence of perceived teacher expectations, teacher support and use of reform practices on mathematics standardised test scores. Our study provides empirical evidence that students’ perceptions that their teachers believe that they are capable of learning and understanding mathematics positively relate to their Mastery and Performance Goal orientations and mathematics Utility. Further, we found an interaction between students’ perceptions of the frequency of reform practice use in their mathematics classroom and mathematics efficacy. For students with lower mathematics efficacy, their perceptions of more frequent use of reform practices in their mathematics class related to higher standardised test performance. For students with higher mathematics efficacy, perceived use of reform practices did not significantly relate to standardised test performance. These data suggest that frequent exposure to reform practices is especially critical for students who judge themselves as being less capable in mathematics and who are lower performers. This finding differs from prior research that has suggested that reform practice use benefits higher achievers but not lower achievers. An important implication of our study is the need for coordinated, ongoing professional development that highlights reform pedagogy and beliefs while also focusing on student motivation theories and strategies.     

Validity of Alternative Cut-Off Scores for the Back-Saver Sit and Reach Test

The purpose of the study was to determine if currently used FITNESSGRAM® cut-off scores for the Back Saver Sit and Reach Test had the best criterion-referenced validity evidence for 6–12 year old children. Secondary analyses of an existing data set focused on the passive straight leg raise and Back Saver Sit and Reach Test flexibility scores of 87 boys and 91 girls. The reliability and validity evidence was determined for Back Saver Sit and Reach Test cut-off scores of 7, 8, 9, and 10 inches for the boys and girls separately. Reliability evidence for passive straight leg raise and all Back Saver Sit and Reach Test cut-off scores was good to excellent. The most valid cut-off scores had the highest proportion of correct decisions associated with sensitivity ≥.67, and false-positive rate ≤.24. Based on moderate validity evidence, the best Back Saver Sit and Reach Test cut-off scores for 6–12 year olds are 8 and 9 inches for boys and girls, respectively. These findings should be cross-validated with other samples.     

Submaximal Treadmill Exercise Test to Predict VO2max in Fit Adults

This study was designed to develop a single-stage submaximal treadmill jogging (TMJ) test to predict VO₂max in fit adults. Participants (N?=?400; men?=?250 and women?=?150), ages 18 to 40 years, successfully completed a maximal graded exercise test (GXT) at 1 of 3 laboratories to determine VO₂max. The TMJ test was completed during the first 2 stages of the GXT. Following 3 min of walking (Stage 1), participants achieved a steady-state heart rate (HR) while exercising at a comfortable self-selected submaximal jogging speed at level grade (Stage 2). Gender, age, body mass, steady-state HR, and jogging speed (mph) were included as independent variables in the following multiple linear regression model to predict VO₂max (R?=?0.91, standard error of estimate [SEE]?=?2.52 mL?·?kg^?1?·?min^?1): VO₂max (mL?·?kg^?1?·?min^?1)?=?58.687?+?(7.520 × Gender; 0?=?woman and 1?=?man)?+?(4.334 × mph) ? (0.211 × kg) ? (0.148 × HR) ? (0.107 × Age). Based on the predicted residual sum of squares (PRESS) statistics (R_PRESS?=?0.91, SEE _PRESS?=?2.54 mL?·?kg^?1?·?min^?1) and small total error (TE; 2.50 mL?·?kg^?1?·?min^?1; 5.3% of VO₂max) and constant error (CE; ?0.008 mL?·?kg^?1?·?min^?1) terms, this new prediction equation displays minimal shrinkage. It should also demonstrate similar accuracy when it is applied to other samples that include participants of comparable age, body mass, and aerobic fitness level. This simple TMJ test and its corresponding regression model provides a relatively safe, convenient, and accurate way to predict VO₂max in fit adults, ages 18 to 40 years.